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15.401

15.401 Finance Theory15.401 Finance TheoryMIT Sloan MBA Program

Andrew W. LoAndrew W. LoHarris & Harris Group Professor, MIT Sloan SchoolHarris & Harris Group Professor, MIT Sloan School

Lectures 4Lectures 4––66: Fixed-Income Securities: Fixed-Income Securities

© 2007–2008 by Andrew W. Lo

© 2007–2008 by Andrew W. LoLectures 4–6: Fixed-Income Securities

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Critical ConceptsCritical ConceptsIndustry OverviewValuationValuation of Discount BondsValuation of Coupon BondsMeasures of Interest-Rate RiskCorporate Bonds and Default RiskThe Sub-Prime Crisis

ReadingsBrealey, Myers, and Allen Chapters 23–25


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Industry OverviewIndustry Overview

Fixed-income securities are financial claims with promised cashflows of known fixed amount paid at fixed dates.

Classification of Fixed-Income Securities:Treasury Securities– U.S. Treasury securities (bills, notes, bonds)– Bunds, JGBs, U.K. Gilts– ….

Federal Agency Securities– Securities issued by federal agencies (FHLB, FNMA $\ldots$)

Corporate Securities– Commercial paper– Medium-term notes (MTNs)– Corporate bonds – ….

Municipal SecuritiesMortgage-Backed SecuritiesDerivatives (CDO’s, CDS’s, etc.)


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U.S. Bond Market Debt 2006 ($Billions)

Municipal, 2,337.50, 9%

Treasury, 4,283.80, 16%

Mortgage-Related,

6,400.40, 24%Corporate,

5,209.70, 19%

Federal Agency,

2,665.20, 10%

Money Markets,

3,818.90, 14%

Asset-Backed, 2,016.70, 8%


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Courtesy of SIFMA. Used with permission. The Securities Industry and Financial Markets Association (SIFMA) prepared this material for informational purposes only. SIFMA obtained this information from multiple sources believed to be reliable as of the date of publication; SIFMA, however, makes no representations as to the accuracy or completeness of such third party information. SIFMA has no obligation to update, modify or amend this information or to otherwise notify a reader thereof in the event that any such information becomes outdated, inaccurate, or incomplete.


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U.S. Bond Market Issuance 2006 ($Billions)

Municipal, 265.3, 6%

Treasury, 599.8, 14%

Mortgage-Related,

1,475.30, 34%

Corporate, 748.7, 17%

Federal Agency, 546.9,

13%

Asset-Backed, 674.6, 16%


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Courtesy of SIFMA.Used with permission. The Securities Industry and Financial Markets Association (SIFMA) prepared this material for informational purposes only. SIFMA obtained this information from multiple sources believed to be reliable as of the date of publication; SIFMA, however, makes no representations as to the accuracy or completeness of such third party information. SIFMA has no obligation to update, modify or amend this information or to otherwise notify a reader thereof in the event that any such information becomes outdated, inaccurate, or incomplete


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Courtesy of SIFMA. Used with permission. The Securities Industry and Financial Markets Association (SIFMA) prepared this material for informational purposes only. SIFMA obtained this information from multiple sources believed to be reliable as of the date of publication; SIFMA, however, makes no representations as to the accuracy or completeness of such third party information. SIFMA has no obligation to update, modify or amend this information or to otherwise notify a reader thereof in the event that any such information becomes outdated, inaccurate, or incomplete.


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Fixed-Income Market Participants

Issuers:GovernmentsCorporationsCommercial BanksStatesMunicipalitiesSPVsForeign Institutions

Intermediaries:Primary DealersOther DealersInvestment BanksCredit-rating AgenciesCredit EnhancersLiquidity Enhancers

Investors:GovernmentsPension FundsInsurance CompaniesCommercial BanksMutual FundsHedge FundsForeign InstitutionsIndividuals


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ValuationValuation

Cashflow:MaturityCouponPrincipal

Example. A 3-year bond with principal of $1,000 and annual coupon payment of 5% has the following cashflow:


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ValuationValuation

Components of ValuationTime value of principal and couponsRisks– Inflation– Credit– Timing (callability)– Liquidity– Currency

For Now, Consider Riskless Debt OnlyU.S. government debt (is it truly riskless?)Consider risky debt later


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Valuation of Discount BondsValuation of Discount Bonds

Pure Discount BondNo coupons, single payment of principal at maturityBond trades at a “discount” to face valueAlso known as zero-coupon bonds or STRIPS*Valuation is straightforward application of NPV

Note: (P0, r, F) is “over-determined”; given two, the third is determined

Now What If r Varies Over Time?Different interest rates from one year to the nextDenote by rt the spot rate of interest in year t

*Separate Trading of Registered Interest and Principal Securities


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If r Varies Over TimeDenote by Rt the one-year spot rate of interest in year t

But we don’t observe the entire sequence of future spot rates today!

Today’s T-year spot rate is an “average” of one-year future spot rates(P0,F,r0,T) is over-determined


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Example:On 20010801, STRIPS are traded at the following prices:

For the 5-year STRIPS, we have


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Suppose We Observe Several Discount Bond Prices Today

Term Structure of Interest Rates


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Term Structure Contain Information About Future Interest Rates

What are the implications of each of the two term structures?

Maturity

r0,t


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Implicit in current bond prices are forecasts of future spot rates!These current forecasts are called one-year forward ratesTo distinguish them from spot rates, we use new notation:


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More Generally:Forward interest rates are today’s rates for transactions between two future dates, for instance, t1 and t2.For a forward transaction to borrow money in the future:– Terms of transaction is agreed on today, t = 0– Loan is received on a future date t1– Repayment of the loan occurs on date t2

Note: future spot rates can be (and usually are) different from current corresponding forward rates


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Valuation of Discount BondsValuation of Discount BondsExample:

As the CFO of a U.S. multinational, you expect to repatriate $10MM from a foreign subsidiary in one year, which will be used to pay dividends one year afterwards. Not knowing the interest rates in one year, you would like to lock into a lending rate one year from now for a period of one year. What should you do? The current interest rates are:

Strategy:Borrow $9.524MM now for one year at 5%Invest the proceeds $9.524MM for two years at 7%


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Valuation of Discount BondsValuation of Discount BondsExample (cont):

Outcome (in millions of dollars):

The locked-in 1-year lending rate one year from now is 9.04%, which is the one-year forward rate for Year 2


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Example:Suppose that discount bond prices are as follows:

A customer would like to have a forward contract to borrow $20MM three years from now for one year. Can you (a bank) quote a rate for this forward loan?

All you need is the forward rate f4 which should be your quote for the forward loan


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Strategy:Buy 20,000,000 of 3 year discount bonds, costing

Finance this by (short)selling 4 year discount bonds of amount

This creates a liability in year 4 in the amount $21,701,403Aside: A shortsales is a particular financial transaction in which an individual can sell a security that s/he does not own by borrowing the security from another party, selling it and receiving the proceeds, and then buying back the security and returning it to the orginal owner at a later date, possibly with a capital gain or loss.


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Cashflows from this strategy (in million dollars):

The yield for this strategy or “synthetic bond return” is given by:


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Valuation of Coupon BondsValuation of Coupon Bonds

Coupon BondsIntermediate payments in addition to final principal paymentCoupon bonds can trade at discounts or premiums to face valueValuation is straightforward application of NPV (how?)

Example:3-year bond of $1,000 par value with 5% coupon


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Valuation of Coupon Bonds

Since future spot rates are unobservable, summarize them with y

y is called the yield-to-maturity of a bondIt is a complex average of all future spot ratesThere is usually no closed-form solution for y; numerical methods must be used to compute it (Tth-degree polynomial)(P0, y, C) is over-determined; any two determines the thirdFor pure discount bonds, the YTM’s are the current spot ratesGraph of coupon-bond y against maturities is called the yield curve


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U.S. Treasury Yield Curves

Source: Bloomberg


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Time Series of U.S. Treasury Security Yields


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Valuation of Coupon BondsValuation of Coupon BondsModels of the Term Structure

Expectations HypothesisLiquidity PreferencePreferred HabitatMarket SegmentationContinuous-Time Models– Vasicek, Cox-Ingersoll-Ross, Heath-Jarrow-Morton

Expectations HypothesisExpected Future Spot = Current Forward

E0[Rk] = fk


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Valuation of Coupon BondsValuation of Coupon BondsLiquidity Preference Model

Investors prefer liquidityLong-term borrowing requires a premiumExpected future spot < current forward

E[Rk] < fk

E[Rk] = fk − Liquidity Premium


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Another Valuation Method for Coupon BondsTheorem: All coupon bonds are portfolios of pure discount bondsValuation of discount bonds implies valuation of coupon bondsProof?

Example:3-Year 5% bondSum of the followingdiscount bonds:– 50 1-Year STRIPS– 50 2-Year STRIPS– 1050 3-Year STRIPS


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Example (cont):Price of 3-Year coupon bond must equal the cost of this portfolioWhat if it does not?

In General:

If this relation is violated, arbitrage opportunities existFor example, suppose that

Short the coupon bond, buy C discount bonds of all maturities up to Tand F discount bonds of maturity TNo risk, positive profits ⇒ arbitrage

P = C P0,1 + C P0,2 + · · · + (C + F )PO,T

P > C P0,1 + C P0,2 + · · · + (C + F )PO,T


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What About Multiple Coupon Bonds?

Suppose n is much bigger than T (more bonds than maturity dates)This system is over-determined: T unknowns, n linear equationsWhat happens if a solution does not exist?This is the basis for fixed-income arbitrage strategies


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Measures of InterestMeasures of Interest--Rate RiskRate Risk

Bonds Subject To Interest-Rate RiskAs interest rates change, bond prices also changeSensitivity of price to changes in yield measures risk


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Measures of InterestMeasures of Interest--Rate RiskRate RiskMacaulay Duration

Weighted average term to maturity

Sensitivity of bond prices to yield changes

Dm =TXk=1

k · ωkqX

k=1

ωk = 1

ωk =Ck/(1 + y)k

P=

PV(Ck)

P

P =TXk=1

Ck(1 + y)k

∂P

∂y=

−11+ y

TXk=1

k · Ck(1 + y)k

1

P

∂P

∂y= − Dm

1+ y= −D∗m Modified Duration


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Measures of InterestMeasures of Interest--Rate RiskRate RiskExample:

Consider a 4-year T-note with face value $100 and 7% coupon, selling at $103.50, yielding 6%.For T-notes, coupons are paid semi-annually. Using 6-month intervals, the coupon rate is 3.5% and the yield is 3%.


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Duration (in 1/2 year units) is

Modified duration (volatility) is

Price risk at y=0.03 is

Note: If the yield moves up by 0.1%, the bond price decreases by 0.6860%

Example (cont):


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Duration decreases with coupon rateDuration decreases with YTMDuration usually increases with maturity– For bonds selling at par or at a premium, duration always increases

with maturity– For deep discount bonds, duration can decrease with maturity– Empirically, duration usually increases with maturity


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For intra-year coupons and annual yield y

ConvexitySensitivity of duration to yield changes

Annual Dm =TXk=1

k · ωk/q

Annual D∗m = Annual Dm/(1 +y

q)

∂2P

∂y2=

1

(1+ y)2

TXk=1

k · (k+1) · Ck(1 + y)k

1

P

∂2P

∂y2= Vm


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Relation between duration and convexity:

Second-order approximation to bond-price functionPortfolio versions:

P (y0) ≈ P (y) +∂P

∂y(y) · (y0 − y) +

∂2P

∂y2(y) · (y

0 − y)22

= P (y) ··1−D∗m(y0 − y) +

1

2Vm(y

0 − y)2¸

P =Xj

Pj

D∗m(P ) ≡ − 1

P

∂P

∂y=

Xj

Pj

PD∗m,j

V ∗m(P ) ≡ − 1

P

∂2P

∂y2=

Xj

Pj

PV ∗m,j


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D∗m =1

1+ 0.062

8Xk=1

kCk

2P (1 + 0.062 )k

= 3.509846

Vm =1

(1+ 0.062 )2

8Xk=1

k(k+1)Ck

4P (1 + 0.062 )k

= 14.805972

P (y0) ≈ P (0.06)³1 − 3.509846(y0− 0.06) +

14.805972(y0− 0.06)2

2

!

P (0.08) ≈ P (0.06)(1− 0.0701969+ 0.0029611)

≈ 93.276427

P (0.08) = 93.267255


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Corporate Bonds and Default RiskCorporate Bonds and Default Risk

Credit Risk Moody's S&P Fitch

Investment GradeHighest Quality Aaa AAA AAAHigh Quality (Very Strong) Aa AA AAUpper Medium Grade (Strong) A A AMedium Grade Baa BBB BBB

Not Investment GradeSomewhat Speculative Ba BB BBSpeculative B B BHighly Speculative Caa CCC CCCMost Speculative Ca CC CCImminent Default C C CDefault C D D

Non-Government Bonds Carry Default RiskA default is when a debt issuer fails to make a promised payment (interest or principal)Credit ratings by rating agencies (e.g., Moody's and S&P) provide indications of the likelihood of default by each issuer.


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Moody’s Baa 10-Year Treasury Yield

Source: Fung and Hsieh (2007)


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Corporate Bonds and Default RiskCorporate Bonds and Default RiskWhat’s In The Premium?

Expected default loss, tax premium, systematic risk premium (Elton, et al 2001)– 17.8% contribution from default on 10-year A-rated industrials

Default, recovery, tax, jumps, liquidity, and market factors (Delianedisand Geske, 2001)– 5-22% contribution from default

Credit risk, illiquidity, call and conversion features, asymmetric tax treatments of corporates and Treasuries (Huang and Huang 2002)– 20-30% contribution from credit risk

Liquidity premium, carrying costs, taxes, or simply pricing errors (Saunders and Allen 2002)


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Corporate Bonds and Default RiskCorporate Bonds and Default RiskDecomposition of Corporate Bond Yields

Promised YTM is the yield if default does not occurExpected YTM is the probability-weighted average of all possible yieldsDefault premium is the difference between promised yield and expected yieldRisk premium (of a bond) is the difference between the expected yield on a risky bond and the yield on a risk-free bond of similar maturity and coupon rate

Example: Suppose all bonds have par value $1,000 and10-year Treasury STRIPS is selling at $463.19, yielding 8%10-year zero issued by XYZ Inc. is selling at $321.97Expected payoff from XYZ's 10-year zero is $762.22


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Corporate Bonds and Default RiskCorporate Bonds and Default RiskFor the 10-year zero issued by XYZ:


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Decomposition of Corporate Bond Yields


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The SubThe Sub--Prime CrisisPrime Crisis

Why Securitize Loans?Repack risks to yield more homogeneity within categoriesMore efficient allocation of riskCreates more risk-bearing capacityProvides greater transparencySupports economic growthBenefits of sub-prime market

But Successful Securitization Requires:DiversificationAccurate risk measurement“Normal” market conditionsReasonably sophisticated investors


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“Confessions of a Risk Manager” in The Economist, August 7, 2008:

Like most banks we owned a portfolio of different tranches of collateralised-debt obligations (CDOs), which are packages of asset-backed securities. Our business and risk strategy was to buy pools of assets, mainly bonds; warehouse them on our own balance-sheet and structure them into CDOs; and finally distribute them to end investors. We were most eager to sell the non-investment-grade tranches, and our risk approvals were conditional on reducing these to zero. We would allow positions of the top-rated AAA and super-senior (even better than AAA) tranches to be held on our own balance-sheet as the default risk was deemed to be well protected by all the lower tranches, whichwould have to absorb any prior losses.

© The Economist. All rights reserved. This content is excluded from our Creative Commons licenseFor more information, see

. http://ocw.mit.edu/fairuse .

http://ocw.mit.edu/fairuse


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“Confessions of a Risk Manager” in The Economist, August 7, 2008:In May 2005 we held AAA tranches, expecting them to rise in value, and sold non-investment-grade tranches, expecting them to go down. From a risk-management point of view, this was perfect: have a long position in the low-risk asset, and a short one in the higher-risk one. But the reverse happened of what we had expected: AAA tranches went down in price and non-investment-grade tranches went up, resulting in losses as we marked the positions to market. This was entirely counter-intuitive. Explanations of why this had happened were confusing and focused on complicated cross-correlations between tranches. In essence it turned out that there had been a short squeeze in non-investment-grade tranches, driving their prices up, and a general selling of all more senior structured tranches, even the very best AAA ones.

© The Economist. All rights reserved. This content is excluded from our Creative Commons license. For more information, see http://ocw.mit.edu/fairuse .

http://ocw.mit.edu/fairuse


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An Illustrative ExampleAn Illustrative Example

Consider Simple Securitization Example:Two identical one-period loans, face value $1,000Loans are risky; they can default with prob. 10%Consider packing them into a portfolioIssue two new claims on this portfolio, S and JLet S have different (higher) priority than JWhat are the properties of S and J?What have we accomplished with this “innovation”?

Let’s Look At The Numbers!


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I.O.U.$1,000

$0 (Default)

90%

10%

Price = 90% × $1,000 + 10% × $0= $900

I.O.U.$1,000

$0 (Default)

90%

10%

Price = 90% × $1,000 + 10% × $0= $900


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I.O.U.$1,000

I.O.U.$1,000

Portfolio

Assuming Independent Defaults

Portfolio Value Prob.

$2,000 81%

$1,000 18%

$0 1%


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I.O.U.$1,000

I.O.U.$1,000

Portfolio

C.D.O.$1,000

C.D.O.$1,000

Senior Tranche

Junior Tranche


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Portfolio Value Prob. Senior

TrancheJunior

Tranche

$1,000 $1,000

$0

$0

$1,000

$0

$2,000 81%

$1,000 18%

$0 1%

Bad StateFor Senior

Tranche (1%)Bad StateFor Junior

Tranche (19%)

C.D.O.$1,000

C.D.O.$1,000

Senior Tranche

Junior Tranche


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“Similar” to Senior

Tranche?

“Similar” to Junior

Tranche?

Non-Investment Grade

Source: Moody’s


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TrancheJunior

Tranche

$1,000 $1,000

$0

$0

$1,000

$0

$2,000 81%

$1,000 18%

$0 1%

C.D.O.$1,000

Senior Tranche

C.D.O.$1,000 Price for Senior Tranche = 99% × $1,000 + 1% × $0

= $990

Price for Junior Tranche = 81% × $1,000 + 19% × $0= $810

Junior Tranche


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C.D.O.$1,000

Senior Tranche

C.D.O.$1,000

Junior Tranche


TrancheJunior

Tranche

$1,000 $1,000

$0

$0

$1,000

$0

$2,000 81%

$1,000 18%

$0 1%

But What If Defaults Become Highly Correlated?


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Assuming Perfectly Correlated Defaults


TrancheJunior

Tranche

$1,000 $1,000

$0$0

$2,000 90%

$0 10%

Bad StateFor Senior

Tranche (10%) Bad StateFor Junior

Tranche (10%)

C.D.O.$1,000

Senior Tranche

C.D.O.$1,000

Junior Tranche


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Assuming Perfectly Correlated Defaults

C.D.O.$1,000

Senior Tranche

C.D.O.$1,000

Junior Tranche


TrancheJunior

Tranche

$1,000 $1,000

$0$0

$2,000 90%

$0 10%

Price for Senior Tranche = 90% × $1,000 + 10% × $0= $900 (was $990)

Price for Junior Tranche = 90% × $1,000 + 10% × $0= $900 (was $810)


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ImplicationsImplicationsTo This Basic Story, Add:

Very low default rates (new securities)Very low correlation of defaults (initially)Aaa for senior tranche (almost riskless)Demand for senior tranche (pension funds)Demand for junior tranche (hedge funds)Fees for origination, rating, leverage, etc.Insurance (monoline, CDS, etc.)Equity bear market, FANNIE, FREDDIE

Then, National Real-Estate Market DeclinesDefault correlation risesSenior tranche declinesJunior tranche increasesRatings declineUnwind ⇒ Losses ⇒ Unwind ⇒ …

C.D.O.$1,000


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Key PointsKey PointsValuation of riskless pure discount bonds using NPV toolsCoupon bonds can be priced from discount bonds via arbitrageCurrent bond prices contain information about future interest ratesSpot rates, forward rates, yield-to-maturity, yield curveInterest-rate risk can be measured by duration and convexityCorporate bonds contain other sources of risk


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Additional ReferencesAdditional ReferencesBrennan, M. and E. Schwartz, 1977, “Savings Bonds, Retractable Bonds and Callable Bonds”, Journal of Financial Economics 5, 67–88.Brown, S. and P. Dybvig, 1986, “The Empirical Implications of the Cox, Ingersoll, Ross Theory of the Term Structure of Interest Rates”, Journal of Finance 41,617–632.Campbell, J., 1986, “A Defense for the Traditional Hypotheses about the Term Structure of Interest Rates”, Journal of Finance 36, 769–800.Cox, J., Ingersoll, J. and S. Ross, 1981, “A Re-examination of Traditional Hypotheses About the Term Structure of Interest Rates”, Journal of Finance 36, 769–799.Cox, J., Ingersoll, J. and S. Ross, 1985, “A Theory of the Term Structure of Interest Rates”, Econometrica 53, 385-–407.Heath, D., Jarrow, R., and A. Morton, 1992, “Bond Pricing and the Term Structure of Interest Rates: A New Methodology for Contingent Claims Valuation”, Econometrica 60, 77-–105.Ho, T. and S. Lee, 1986, “Term Structure Movements and Pricing Interest Rate Contingent Claims'', Journal of Finance41, 1011–1029.Jegadeesh, N. and B. Tuckman, eds., 2000, Advanced Fixed-Income Valuation Tools. New York: John Wiley & Sons.McCulloch, H., 1990, “U.S. Government Term Structure Data”, Appendix to R. Shiller, “The Term Structure of Interest Rates”, in Benjamin M. Friedman and Frank H. Hahn eds. Handbook of Monetary Economics. Amsterdam: North-Holland.Sundaresan, S., 1997, Fixed Income Markets and Their Derivatives. Cincinnati, OH: South-Western College Publishing.Tuckman, B., 1995, Fixed Income Securities: Tools for Today's Markets. New York: John Wiley & Sons.Vasicek, O., 1977, “An Equilibrium Characterization of the Term Structure”, Journal of Financial Economics 5, 177–188.

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